In robust control theory, the central problem is the design of controllers for dynamic physical systems so as to guarantee stability and performance in the face of large modeling uncertainties and worst case inputs. The central problem for the design of robust controllers for a given plant under mixed parametric uncertainty remains the structured stability analysis problem.

The numerical range is a well established theory that has been studied for years and has many applications in mathematics and engineering. The numerical range of an n x n matrix, also known as its field of values, is reformulated as the image of a smooth quadratic mapping from the n -1 dimensional complex projective space to the complex plane.  For robust control analysis, it is the generalized numerical range that is of importance.  From the fundamental observation that the return difference map of a diagonally perturbed system is a holomorphic function of several complex variables -- and hence an open mapping -- a self-contained approach to the generalized numerical range computation of the structured stability margin is derived.

Next, differential topological tools are used to investigate the issue of convexity of the generalized numerical range. Convexity of the outer boundary is shown to be equivalent to the Morse function property of a family of quadratic forms defined over the complex projective space and parameterized by a sphere.

Finally, a differential topology explanation of the lack of convexity for 4 blocks is provided. The fundamental issue that underlies the dimension-dependence of the structured stability margin is the fact that a family of quadratic forms defined over complex projective space indexed by S^m-1 can have the Morse property for all values of the parameter only for m < 4

Another explanation of the dimension-dependence is the real geometric feature that the discriminant of a Hermitian matrix is the sum of a varying number of squares depending on the multiplicity of the eigenvalues.

Full thesis in postscript format is available here.